Find the following limit
$$\lim_{x \to 0}\frac{[x]}{x}$$
Where $[x]$ is the greatest integer function.
I tried using the squeeze theorem on this but couldn’t come up with the appropriate functions, can someone help me out?
Edit: I can now see that this limit will not exist.
But what can we say about
$$\lim_{x \to 0}x[1/x]$$
Where again [x] is the greatest integer function, how can i use squeeze theorem to find this?
Best Answer
Defining $u={1\over x}$ we must obtain two limits$$\lim_{u\to \infty}{\lfloor u\rfloor\over u}$$and $$\lim_{u\to -\infty}{\lfloor u\rfloor\over u}$$for $u\to \infty$ we can write$${\lfloor u\rfloor\over \lfloor u\rfloor+1}< {\lfloor u\rfloor\over u}\le1$$and for $u\to -\infty$$$1\le {\lfloor u\rfloor\over u}<{\lfloor u\rfloor\over \lfloor u\rfloor+1}$$by Squeeze theorem, both tend to $1$ and so does $\lim_{x\to 0}x\lfloor{1\over x}\rfloor$